Networked Slepian –Wolf: Theory, Algorithms, and Scaling Laws

1. Networked Slepian–Wolf: Theory, Algorithms, and Scaling Laws R˘azvanCristescu, Member, IEEE, BaltasarBeferull-Lozano, Member, IEEE, Martin Vetterli, Fellow, IEEE IEEE Transactions on Information Theory, Dec., 2005

2. Outline • Introduction • Slepian–Wolf Coding • Problem Formulation • Single Sink Case • Multiple Sink Case • Single Sink Data Gathering • Multiple Sink Data Gathering • Heuristic Approximation Algorithms • Numerical Simulations • Conclusion

3. Introduction • Independent encoding/decoding • Low coding gain • Optimal transmission structure: Shortest path tree • Encoding with explicit communication • Nodes can exploit the data correlation only when the data of other nodes is locally at them). • Without knowing the correlation among nodes a priori. • Distributed source coding: Slepian–Wolf coding • Allow nodes to use joint coding of correlated data without explicit communication • Assume a prior knowledge of global network structure and correlation structure is availlable • Exploiting data correlation without explicit communication (coding at each node Independent ly) • Node can exploit data correlation among nodes without explicit communication. • Optimal transmission structure: Shortest path tree

4. Slepian–Wolf coding

5. Slepian–Wolf coding

6. Slepian–Wolf coding

7. Slepian–Wolf coding

8. Problem Multiple Sink Case Single Sink Case Assume the Slepian–Wolf coding is used. Then, Find a rate allocation that minimizes the total network cost. (2) Find an optimal transmission structure.

9. Preposition • Proposition 1: Separation of source coding and transmission structure optimization.

10. Single-Sink Data Gathering • Optimal Transmission Structure: • Shortest Path Tree

11. Single-Sink Data Gathering Optimization problem Rate Allocation

12. Proof Consider that with weights Since Thus, assigning Yields optimal

13. Rate Allocation R1: the smallest R1: the largest

14. Example

15. Multiple Sink Case • For Node X3, the optimal transmission structure is the minimum-weight tree rooted at X3 and span the sinks S1 and S2. the minimum Steiner tree (NP-complete)

16. Steiner Tree • Euclidean Steiner tree problem • Given N points in the plane, it is required to connect them by lines of minimum total length in such a way that any two points may be interconnected by line segments either directly or via other points and line segments.

17. Steiner Tree • Steiner tree in graphs • Given a weighted graph G(V, E, w) and a subset of its vertices S  V, find a tree of minimal weight which includes all vertices in S. 5 Terminal 6 5 2 2 Steiner points 2 3 3 4 2 2 4 13

18. The Minimum Steiner Tree

19. Existing Approximation • If the weights of the graph are the Euclidean distances, • the Euclidean Steiner tree problem • The existing approximation PTAS [3], with approximation ratio (1+),  > 0.

20. Proposed Heuristic Approximation Algorithms Assumption : Nodes that are outside k-hop neighborhood count very little, in terms of rate, in the local entropy conditioning,

21. Numerical Simulations • Source model: multivariate Gaussian random field. • Correlation model: an exponential model that decays exponentially with the distance between the nodes.

22. Numerical Simulations

23. Numerical Simulations

24. Conclusions • This paper addressed the problem of joint rate allocation and transmission structure optimization for sensor networks. • It was shown that • in single-sink case the optimal transmission structure is the shortest path tree. • in the multiple-sink case the optimization of transmission structure is NP-complete. • Steiner tree problem